Discovery of mathematical descriptors of physical phenomena from observational and simulated data, as opposed to from the first principles, is a rapidly evolving research area. Two factors, time-dependence of the inputs and hidden translation invariance, are known to complicate this task. To ameliorate these challenges, we combine Lagrangian dynamic mode decomposition with a locally time-invariant approximation of the Koopman operator. The former component of our method yields the best linear estimator of the system's dynamics, while the latter deals with the system's nonlinearity and non-autonomous behavior. We provide theoretical estimators (bounds) of prediction accuracy and perturbation error to guide the selection of both rank truncation and temporal discretization. We demonstrate the performance of our approach on several non-autonomous problems, including two-dimensional Navier-Stokes equations.

翻译：从观测与模拟数据（而非第一性原理）中揭示物理现象的数学描述子，是一个快速发展的研究领域。已知有两类因素会复杂化这一任务：输入的时间依赖性以及隐藏的平移不变性。为应对这些挑战，我们结合了拉格朗日动态模态分解与库普曼算子的局部时间不变近似。该方法的前一部分提供系统动力学的最优线性估计器，后一部分则处理系统的非线性和非自治行为。我们提供了预测精度和扰动误差的理论估计（界），以指导秩截断与时间离散化的选择。我们通过多个非自治问题（包括二维纳维-斯托克斯方程）验证了方法的性能。